Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,14,16)(17,18)(19,28)(20,22)(21,26)(23,25)(24,30)(27,31)(29,32)(33,34,39,38)(35,43,46,40)(36,41,45,47)(37,42,44,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,14)(15,16)(17,19,23,30)(18,24,25,28)(20,29,27,26)(21,31,32,22)(33,35)(34,40)(36,44)(37,45)(38,43)(39,46)(41,42)(47,48);(3,11)(7,8)(12,14)(13,16)(18,25)(20,27)(24,28)(26,29)(33,36,39,45)(34,41,38,47)(35,44,46,37)(40,42,43,48);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,20,23,27)(18,22,25,31)(19,29,30,26)(21,28,32,24)(33,36)(34,41)(35,44)(37,46)(38,47)(39,45)(40,42)(43,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,16)(10,15)(11,14)(17,21,23,32)(18,26,25,29)(19,22,30,31)(20,24,27,28)(33,37,39,44)(34,42,38,48)(35,45,46,36)(40,41,43,47);(1,5)(2,9)(3,13)(4,10)(6,15)(7,14)(8,12)(11,16)(17,22)(18,27)(19,21)(20,25)(23,31)(24,26)(28,29)(30,32)(33,38,39,34)(35,43,46,40)(36,47,45,41)(37,42,44,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9080356);